Collaborative filtering¶
A distinction is often made between two forms of data collection for recommendation systems. Explicit feedback relies on the user giving explicit signals about their preferences i.e. review ratings. Where as, implicit feedback refers to non-explicit signals of preference e.g. user watch-time. Traditionally, recommender systems can be split into three types:
Collaborative filtering (CF): CF produces recommendations based on the knowledge of users’ attitudes towards items, that is, it uses the “wisdom of the crowd” to recommend items.
Content-based (CB): CB recommender systems focus on the attributes of the items to recommend other items similar to what the user likes, based on their previous actions or explicit feedback.
Hybrid recommendation systems: Hybrid methods are a combination of CB recommending and CF methods
In many applications, content-based features are not easy to extract, and thus, collaborative filtering approaches are preferred. Thus, we will only explore collaborative filtering methods from now on.
CF methods typically fall into three types, memory-based, model-based and more recently deep-learning based (Su & Khoshgoftaar, 2009, He et al., 2017). Neighbour-based CF and item-based/user-based top-N recommendations are typical examples of memory-based systems that utilises user rating data to compute the similarity between users or items. As mentioned previously, common model-based approaches include Bayesian networks, latent semantic models and markov decision processes. In this investigation, we will utilise a weighted matrix factorization approach. Later on, we will generalize the matrix factorization algorithm via a non-linear neural architecture (a softmax model).
However, there are a number of limitations to our approaches such as the inability to model the order of interactions. For instance, Markov chain algorithms (Rendle et al., 2010) can not only encode the same information as traditional CF methods but also the order in which user’s interacted with the items. Furthermore, the sparsity of the frequency matrix (described later on), makes computations prohibitly expensive in real-world settings, without some optimization.
Quick Links:¶
Setup¶
The next few code cells details the initial preparatory steps needed for the development of our collaborative filtering models, namely importing the required libraries; scaling the ids of users and artists;constructing a indicator variable for presence of user-artist interaction;finding the most assigned tag of an artist.
from __future__ import print_function
import numpy as np
import pandas as pd
import collections
from IPython import display
from matplotlib import pyplot as plt
import sklearn
import sklearn.manifold
import tensorflow.compat.v1 as tf
tf.disable_v2_behavior()
tf.logging.set_verbosity(tf.logging.ERROR)
# Add some convenience functions to Pandas DataFrame.
pd.options.display.max_rows = 10
pd.options.display.float_format = '{:.3f}'.format
# Install Altair and activate its colab renderer.
print("Installing Altair...")
!pip install git+git://github.com/altair-viz/altair.git
import altair as alt
alt.data_transformers.enable('default', max_rows=None)
alt.renderers.enable('colab')
print("Done installing Altair.")
2021-11-28 09:20:08.305373: W tensorflow/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libcudart.so.11.0'; dlerror: libcudart.so.11.0: cannot open shared object file: No such file or directory
2021-11-28 09:20:08.305412: I tensorflow/stream_executor/cuda/cudart_stub.cc:29] Ignore above cudart dlerror if you do not have a GPU set up on your machine.
WARNING:tensorflow:From /opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/tensorflow/python/compat/v2_compat.py:111: disable_resource_variables (from tensorflow.python.ops.variable_scope) is deprecated and will be removed in a future version.
Instructions for updating:
non-resource variables are not supported in the long term
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# NEEDED FOR GOOGLE COLAB
# from google.colab import auth
#from google.colab import drive
# import gspread
# from oauth2client.client import GoogleCredentials
# drive.mount('/content/drive/')
# os.chdir("/content/drive/My Drive/DCU/fouth_year/advanced_machine_learning/music-recommodation-system")
Helper functions
def calculate_sparsity(M):
"""
Computes sparsity of frequency matrix
"""
matrix_size = len((M['userID'].unique())) * len((M['artistID'].unique())) # Number of possible interactions in the matrix
num_plays = len(M['weight']) # Number of weights
sparsity = (float(num_plays/matrix_size))
return sparsity
def build_music_sparse_tensor(music_df):
"""
Args:
ratings_df: a pd.DataFrame with `userID`, `artistID` and `weight` columns.
num_rows: an integer representing the number of rows in the frequency matrix
num_rows: an integer representing the number of columns in the frequency matrix
Returns:
a tf.SparseTensor representing the feedback matrix.
"""
indices = music_df[['userID', 'artistID']].values
values = music_df['weight'].values
return tf.SparseTensor(
indices=indices,
values=values,
dense_shape=[num_users, num_artist])
def preproces_ids(music_df):
"""
Args:
ratings_df: a pd.DataFrame with `userID`, `artistID` and `weight` columns.
Returns:
a pd.DataFrame where userIDs and artistIDs now start at 1
and end at n and m (defined above), respectively
two dictionary preserving the orginal ids.
"""
unique_user_ids_list = sorted(music_df['userID'].unique())
print(unique_user_ids_list[0])
unique_user_ids = dict(zip(range(0, len(unique_user_ids_list) ),unique_user_ids_list))
unique_user_ids_switched = dict(zip(unique_user_ids_list, range(0, len(unique_user_ids) )))
unique_artist_ids_list = sorted(music_df['artistID'].unique())
unique_artist_ids = dict(zip(range(0, len(unique_artist_ids_list) ),unique_artist_ids_list))
unique_artist_ids_switched = dict(zip(unique_artist_ids_list, range(0, len(unique_artist_ids_list) )))
music_df['userID'] = music_df['userID'].map(unique_user_ids_switched)
music_df['artistID'] = music_df['artistID'].map(unique_artist_ids_switched)
return music_df, unique_user_ids, unique_artist_ids
def split_dataframe(df, holdout_fraction=0.1):
"""Splits a DataFrame into training and test sets.
Args:
df: a dataframe.
holdout_fraction: fraction of dataframe rows to use in the test set.
Returns:
train: dataframe for training
test: dataframe for testing
"""
test = df.sample(frac=holdout_fraction, replace=False)
train = df[~df.index.isin(test.index)]
return train, test
Traditional recommender system development relies on explicit feedback. Many models were designed to tackle this issue as a regression problem. For instance, the input of the model would be a matrix \(F_{nm}\) denoting user’s (m) preference of items (n) on a scale. In the classic movie ratings example, this preference would be users giving a 1-to-5 star rating to different movies.
This dataset contains implicit feedback: that is, observed logs of user interactions with items, in this instance user’s listening counts to artists. However, implicit feedback does not signal negativity, in the same way as a 1-star rating would. In our data, a user could listen to song of an artist a limited number of times. But that does not necessarily mean that the particular user has an aversion to that artist i.e. it could be part of a curated playlist by another user. Therefore, we decide to construct a binary matrix, which has a value of one if the observation is observed (i.e. a listening count has been logged between an artist and a user). Note, a 0 is not used to describe unobserved artist-user interactions. This is for optimization reasons, explained below.
user_artists = pd.read_csv('data/user_artists.dat', sep='\t')
user_artists['weight'] = 1
artists = pd.read_csv('data/artists.dat', sep='\t')
artists.rename({'id':'artistID'}, inplace=True, axis=1)
user_taggedartists = pd.read_csv(r'data/user_taggedartists-timestamps.dat', sep='\t')
user_taggedartists_years = pd.read_csv(r'data/user_taggedartists.dat', sep='\t')
tags = pd.read_csv(open('data/tags.dat', errors='replace'), sep='\t')
user_taggedartists = pd.merge(user_taggedartists, tags, on=['tagID'])
num_users = user_artists.userID.nunique()
num_artist = artists.artistID.nunique()
collab_filter_df = user_artists
Here, we calculate the top 10 tags by popularity. Then, we assign it to a artist, if the artist has a top 10 tag. If an artist’s tags are not in the top 10, we input ‘N/A’. Note, the next cell can take several mintues to compute.
top_10_tags = user_taggedartists['tagValue'].value_counts().index[0:10]
user_taggedartists['top10TagValue'] = None
for index, row in user_taggedartists.iterrows():
if row['tagValue'] in top_10_tags:
user_taggedartists.iloc[index, -1] = row['tagValue']
user_taggedartists.fillna('N/A',inplace=True)
artists = pd.merge(user_taggedartists, artists, on=['artistID'], how='right')[['artistID','name','top10TagValue','tagValue']].fillna('N/A')
artists.groupby(['artistID','name','top10TagValue']).agg(lambda x:x.value_counts().index[0]).reset_index()
artists = artists.drop_duplicates(subset=['artistID'])
assert artists.artistID.nunique() == num_artist
artists.rename({'tagValue':'mostCommonGenre'},axis=1, inplace=True)
We require two matrices or embeddings to compute a similarity measure (one for quires and one for items), but how do we get these two embeddings?
Matrix Factorisation¶
Figure 2: Data flow chart
First, we need to contsruct the feedback matrix \(F \in R^{m \times n}\), where \(m\) is the number of users and \(n\) is the number of artists. The goal is to two generate two lower-dimensional matrices \(U_{mp}\) and \(V_{np}\) ( with \(p << m\) and \(p << n\)), representing latent user and artist components, so that: $\( F \approx UV^\top \)$
First,we attempt to build the frequency matrix for both training and testing data. tf.SparseTensor is used
for efficient representation. Three separate arguments are used to represent a tensor, namely indices, values, dense_shape, where a value \(A_{ij} = a\) is encoded by setting indices[k] = [i, j] and values[k] = a. The last tensor dense_shape is used to specify the shape of the full underlying matrix. Note, as the indices arguments represent row and columns indices, some pre-processing needs to be performed on artist and user IDs. The IDs should start from 0 and end at \(m-1\) and \(n-1\) for users and artists respectively. Presently, userIDs start at 2. Two dictionaries, orginal_artist_ids, orginal_user_ids will preserve the original ids for analysis purposes later on. Assertions and print statements are used to ensure the validity of the transformations.
colab_filter_df, orginal_user_ids, orginal_artist_ids = preproces_ids(collab_filter_df)
2
colab_filter_df.describe()
| userID | artistID | weight | |
|---|---|---|---|
| count | 92834.000 | 92834.000 | 92834.000 |
| mean | 944.222 | 3235.737 | 1.000 |
| std | 546.751 | 4197.217 | 0.000 |
| min | 0.000 | 0.000 | 1.000 |
| 25% | 470.000 | 430.000 | 1.000 |
| 50% | 944.000 | 1237.000 | 1.000 |
| 75% | 1416.000 | 4266.000 | 1.000 |
| max | 1891.000 | 17631.000 | 1.000 |
Next, we caulcate the number of unique artists, userids and sparisty of our proposed frequency matrix, before splitting into training and test subsets. Quite a sparse matrix indeed!
print(f'Number of unqiue users are: {collab_filter_df["userID"].nunique()}')
print(f'Number of unqiue artists are: {collab_filter_df["artistID"].nunique()}')
print(f'Sparsity of our frequency matrix: {calculate_sparsity(collab_filter_df)}')
Number of unqiue users are: 1892
Number of unqiue artists are: 17632
Sparsity of our frequency matrix: 0.002782815119924182
collab_filter_df.to_csv('data/test_user_artists.csv',index=False)
frequency_m_train, frequency_m_test = split_dataframe(colab_filter_df)
frequency_m_train_tensor = build_music_sparse_tensor(frequency_m_train)
frequency_m_test_tensor = build_music_sparse_tensor(frequency_m_test)
assert num_users == frequency_m_train_tensor.shape.as_list()[0]
assert num_artist == frequency_m_train_tensor.shape.as_list()[1]
assert num_users == frequency_m_test_tensor.shape.as_list()[0]
assert num_artist == frequency_m_test_tensor.shape.as_list()[1]
Training a Matrix factorization model¶
Per the definition above, \(UV^\top\) approximates \(F\). The Mean Squared Error is used to measure this approximation error. In the notation below, k is used to represent the set of observed listening counts, and K is the number of observed listening counts.
However, rather than computing the full prediction matrix, \(UV^\top\) and gathering the entries in the embeddings (corresponding to the observed listening counts) , we only gather the embeddings of the observers pairs and compute their dot products. Thereby, we reduce the complexity from \(O(NM)\) to \(O(Kp)\) where \(p\) is the embedding dimension. Stochastic gradient descent (SGD) is used to minimize the loss (objective) function. The SDG algorithim cycles through the observed listening binary and caulates the prediction according to the following equation.
Then it updates the user and artist as embeddings as shown in the following equations.
where \(\alpha\) denotes the learning rate. The algorithim continues untill convergence is found.
Other matrix factorization algorithms functions are also commonly used such as Alternating Least Squares (Takács and Tikk, 2012). A modified version of the aforementioned function known as Weighted Alternating Least Squares (WALS) is slower than SDG but can be parallelised. For the purposes of this investigation, we are not particularly concerned with training times/latency requirements so we proceed with SDG.
We also decide to add regularization to our model, to avoid overfitting. Overfitting occurs when the model tries to fit the training dataset to hard and does not generalize well to unseen or future data. In the context of artist recommendation, fitting the observed listening counts often emphasizes learning high similarity (between artists with many listeners), but a good embedding representation also requires learning low similarity (between artists with few listeners).
First, we define the two classes train_matrix_norm and build_matrix_norm class. The build_matrix_norm class computes the necessary pre-processing steps before we train the model such as specifying the loss metric to optimise and the loss components( e.g. gravity loss for the regularized model) and the initial artist and user embeddings. train_matrix_norm simply trains the models and outputs figures detailing the the loss metrics and components. The methods build_vanilla() and build_reg_model() computes the necessary pre-processing steps for the non-regularized and regularized model.
### Training a Matrix Factorization model
class train_matrix_norm(object):
"""Simple class that represents a matrix normalisation model"""
def __init__(self, embedding_vars, loss, metrics=None):
"""Initializes a Matrix normalisation model
Args:
embedding_vars: A dictionary of tf.Variables.
loss: A float Tensor. The loss to optimize.
metrics: optional list of dictionaries of Tensors. The metrics in each
dictionary will be plotted in a separate figure during training.
"""
self._embedding_vars = embedding_vars
self._loss = loss
self._metrics = metrics
self._embeddings = {k: None for k in embedding_vars}
self._session = None
@property
def embeddings(self):
"""The embeddings dictionary."""
return self._embeddings
def train(self, num_iterations=100, learning_rate=1.0, plot_results=True,
optimizer=tf.train.GradientDescentOptimizer):
"""Trains the model.
Args:
iterations: number of iterations to run.
learning_rate: optimizer learning rate.
plot_results: whether to plot the results at the end of training.
optimizer: the optimizer to use. Default to SDG
Returns:
The metrics dictionary evaluated at the last iteration.
"""
with self._loss.graph.as_default():
opt = optimizer(learning_rate)
train_op = opt.minimize(self._loss)
local_init_op = tf.group(
tf.variables_initializer(opt.variables()),
tf.local_variables_initializer())
if self._session is None:
self._session = tf.Session()
with self._session.as_default():
self._session.run(tf.global_variables_initializer())
self._session.run(tf.tables_initializer())
tf.train.start_queue_runners()
with self._session.as_default():
local_init_op.run()
iterations = []
metrics = self._metrics or ({},)
metrics_vals = [collections.defaultdict(list) for _ in self._metrics]
# Train and append results.
for i in range(num_iterations + 1):
_, results = self._session.run((train_op, metrics))
if (i % 10 == 0) or i == num_iterations:
print("\r iteration %d: " % i + ", ".join(
["%s=%f" % (k, v) for r in results for k, v in r.items()]),
end='')
iterations.append(i)
for metric_val, result in zip(metrics_vals, results):
for k, v in result.items():
metric_val[k].append(v)
for k, v in self._embedding_vars.items():
self._embeddings[k] = v.eval()
if plot_results:
# Plot the metrics.
num_subplots = len(metrics)+1
fig = plt.figure()
fig.set_size_inches(num_subplots*10, 8)
for i, metric_vals in enumerate(metrics_vals):
ax = fig.add_subplot(1, num_subplots, i+1)
for k, v in metric_vals.items():
ax.plot(iterations, v, label=k)
ax.set_xlim([1, num_iterations])
ax.legend()
return results
class build_matrix_norm():
"""Simple class that represents a matrix normalisation model"""
def __init__(self, listens, embedding_dim=3, regularization_coeff=.1, gravity_coeff=1.,
init_stddev=0.1):
"""Initializes a Matrix normalisation model
Args:
listens: the DataFrame of artist listening counts.
embedding_dim: The dimension of the embedding space.
regularization_coeff: The regularization coefficient lambda.
gravity_coeff: The gravity regularization coefficient lambda_g.
Returns:
A train_matrix_norm object that uses a regularized loss.
"""
self._embedding_vars = embedding_vars
self._loss = loss
self._metrics = metrics
self._embeddings = {k: None for k in embedding_vars}
self._session = None
def sparse_mean_square_error(sparse_listens, user_embeddings, artist_embeddings):
"""
Args:
sparse_listens: A SparseTensor rating matrix, of dense_shape [N, M]
user_embeddings: A dense Tensor U of shape [N, k] where k is the embedding
dimension, such that U_i is the embedding of user i.
artist_embeddings: A dense Tensor V of shape [M, k] where k is the embedding
dimension, such that V_j is the embedding of movie j.
Returns:
A scalar Tensor representing the MSE between the true ratings and the
model's predictions.
"""
predictions = tf.gather_nd(
tf.matmul(user_embeddings, artist_embeddings, transpose_b=True),
sparse_listens.indices)
loss = tf.losses.mean_squared_error(sparse_listens.values, predictions)
return loss
def gravity(U, V):
"""Creates a gravity loss given two embedding matrices."""
return 1. / (U.shape[0].value*V.shape[0].value) * tf.reduce_sum(
tf.matmul(U, U, transpose_a=True) * tf.matmul(V, V, transpose_a=True))
def build_vanilla(embedding_dim=3, init_stddev=1.):
"""performs the necessary preprocessing steps for the regularized model. """
# Initialize the embeddings using a normal distribution.
U = tf.Variable(tf.random.normal(
[frequency_m_train_tensor.dense_shape[0], embedding_dim], stddev=init_stddev))
V = tf.Variable(tf.random.normal(
[frequency_m_train_tensor.dense_shape[1], embedding_dim], stddev=init_stddev))
embeddings = {"userID": U, "artistID": V}
error_train = build_matrix_norm.sparse_mean_square_error(frequency_m_train_tensor, U, V)
error_test = build_matrix_norm.sparse_mean_square_error(frequency_m_test_tensor, U, V)
metrics = {
'train_error': error_train,
'test_error': error_test
}
return train_matrix_norm(embeddings, error_train, [metrics])
def build_reg_model(embedding_dim=3, regularization_coeff=.1, gravity_coeff=1.,
init_stddev=0.1
):
"""performs the necessary preprocessing steps for the regularized model. """
U = tf.Variable(tf.random.normal(
[frequency_m_train_tensor.dense_shape[0], embedding_dim], stddev=init_stddev))
V = tf.Variable(tf.random.normal(
[frequency_m_train_tensor.dense_shape[1], embedding_dim], stddev=init_stddev))
embeddings = {"userID": U, "artistID": V}
error_train = build_matrix_norm.sparse_mean_square_error(frequency_m_train_tensor, U, V)
error_test = build_matrix_norm.sparse_mean_square_error(frequency_m_test_tensor, U, V)
gravity_loss = gravity_coeff * build_matrix_norm.gravity(U, V)
regularization_loss = regularization_coeff * (
tf.reduce_sum(U*U)/U.shape[0].value + tf.reduce_sum(V*V)/V.shape[0].value)
total_loss = error_train + regularization_loss + gravity_loss
losses = {
'train_error_observed': error_train,
'test_error_observed': error_test,
}
loss_components = {
'observed_loss': error_train,
'regularization_loss': regularization_loss,
'gravity_loss': gravity_loss,
}
#embeddings = {"userID": U, "artistID": V}
return train_matrix_norm(embeddings, total_loss, [losses, loss_components])
Vanilla Model (non-regularized)¶
vanilla_model = build_matrix_norm.build_vanilla(embedding_dim=35,init_stddev=.05)
vanilla_model.train(num_iterations=2000, learning_rate=20.)
2021-11-28 09:22:11.997572: W tensorflow/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libcuda.so.1'; dlerror: libcuda.so.1: cannot open shared object file: No such file or directory
2021-11-28 09:22:11.997609: W tensorflow/stream_executor/cuda/cuda_driver.cc:269] failed call to cuInit: UNKNOWN ERROR (303)
2021-11-28 09:22:11.997633: I tensorflow/stream_executor/cuda/cuda_diagnostics.cc:156] kernel driver does not appear to be running on this host (fv-az77-612): /proc/driver/nvidia/version does not exist
2021-11-28 09:22:11.997906: I tensorflow/core/platform/cpu_feature_guard.cc:151] This TensorFlow binary is optimized with oneAPI Deep Neural Network Library (oneDNN) to use the following CPU instructions in performance-critical operations: AVX2 AVX512F FMA
To enable them in other operations, rebuild TensorFlow with the appropriate compiler flags.
iteration 0: train_error=1.000189, test_error=1.000171
iteration 10: train_error=0.998505, test_error=1.000141
iteration 20: train_error=0.996779, test_error=1.000079
iteration 30: train_error=0.994916, test_error=0.999910
iteration 40: train_error=0.992783, test_error=0.999520
iteration 50: train_error=0.990151, test_error=0.998709
iteration 60: train_error=0.986620, test_error=0.997111
iteration 70: train_error=0.981474, test_error=0.994062
iteration 80: train_error=0.973477, test_error=0.988392
iteration 90: train_error=0.960665, test_error=0.978213
iteration 100: train_error=0.940460, test_error=0.960987
iteration 110: train_error=0.910765, test_error=0.934515
iteration 120: train_error=0.872074, test_error=0.898992
iteration 130: train_error=0.828056, test_error=0.857810
iteration 140: train_error=0.782482, test_error=0.814760
iteration 150: train_error=0.736969, test_error=0.771548
iteration 160: train_error=0.692321, test_error=0.728894
iteration 170: train_error=0.649599, test_error=0.687791
iteration 180: train_error=0.609705, test_error=0.649219
iteration 190: train_error=0.573026, test_error=0.613708
iteration 200: train_error=0.539542, test_error=0.581351
iteration 210: train_error=0.509036, test_error=0.551995
iteration 220: train_error=0.481225, test_error=0.525389
iteration 230: train_error=0.455822, test_error=0.501259
iteration 240: train_error=0.432552, test_error=0.479336
iteration 250: train_error=0.411168, test_error=0.459374
iteration 260: train_error=0.391452, test_error=0.441149
iteration 270: train_error=0.373208, test_error=0.424460
iteration 280: train_error=0.356268, test_error=0.409130
iteration 290: train_error=0.340484, test_error=0.395005
iteration 300: train_error=0.325731, test_error=0.381950
iteration 310: train_error=0.311900, test_error=0.369848
iteration 320: train_error=0.298897, test_error=0.358599
iteration 330: train_error=0.286643, test_error=0.348118
iteration 340: train_error=0.275068, test_error=0.338331
iteration 350: train_error=0.264114, test_error=0.329174
iteration 360: train_error=0.253728, test_error=0.320592
iteration 370: train_error=0.243865, test_error=0.312537
iteration 380: train_error=0.234487, test_error=0.304966
iteration 390: train_error=0.225558, test_error=0.297842
iteration 400: train_error=0.217047, test_error=0.291131
iteration 410: train_error=0.208926, test_error=0.284804
iteration 420: train_error=0.201170, test_error=0.278833
iteration 430: train_error=0.193754, test_error=0.273193
iteration 440: train_error=0.186657, test_error=0.267862
iteration 450: train_error=0.179859, test_error=0.262817
iteration 460: train_error=0.173342, test_error=0.258041
iteration 470: train_error=0.167087, test_error=0.253515
iteration 480: train_error=0.161081, test_error=0.249222
iteration 490: train_error=0.155307, test_error=0.245147
iteration 500: train_error=0.149753, test_error=0.241276
iteration 510: train_error=0.144407, test_error=0.237596
iteration 520: train_error=0.139256, test_error=0.234094
iteration 530: train_error=0.134291, test_error=0.230761
iteration 540: train_error=0.129502, test_error=0.227584
iteration 550: train_error=0.124880, test_error=0.224556
iteration 560: train_error=0.120419, test_error=0.221666
iteration 570: train_error=0.116109, test_error=0.218908
iteration 580: train_error=0.111946, test_error=0.216273
iteration 590: train_error=0.107923, test_error=0.213754
iteration 600: train_error=0.104034, test_error=0.211346
iteration 610: train_error=0.100275, test_error=0.209041
iteration 620: train_error=0.096640, test_error=0.206835
iteration 630: train_error=0.093126, test_error=0.204723
iteration 640: train_error=0.089728, test_error=0.202699
iteration 650: train_error=0.086443, test_error=0.200759
iteration 660: train_error=0.083268, test_error=0.198899
iteration 670: train_error=0.080198, test_error=0.197114
iteration 680: train_error=0.077232, test_error=0.195403
iteration 690: train_error=0.074366, test_error=0.193760
iteration 700: train_error=0.071597, test_error=0.192182
iteration 710: train_error=0.068923, test_error=0.190667
iteration 720: train_error=0.066342, test_error=0.189212
iteration 730: train_error=0.063851, test_error=0.187814
iteration 740: train_error=0.061447, test_error=0.186470
iteration 750: train_error=0.059129, test_error=0.185178
iteration 760: train_error=0.056893, test_error=0.183936
iteration 770: train_error=0.054739, test_error=0.182741
iteration 780: train_error=0.052662, test_error=0.181592
iteration 790: train_error=0.050663, test_error=0.180487
iteration 800: train_error=0.048738, test_error=0.179423
iteration 810: train_error=0.046885, test_error=0.178399
iteration 820: train_error=0.045102, test_error=0.177413
iteration 830: train_error=0.043387, test_error=0.176464
iteration 840: train_error=0.041738, test_error=0.175550
iteration 850: train_error=0.040152, test_error=0.174669
iteration 860: train_error=0.038629, test_error=0.173821
iteration 870: train_error=0.037165, test_error=0.173004
iteration 880: train_error=0.035760, test_error=0.172216
iteration 890: train_error=0.034409, test_error=0.171457
iteration 900: train_error=0.033113, test_error=0.170725
iteration 910: train_error=0.031869, test_error=0.170019
iteration 920: train_error=0.030675, test_error=0.169339
iteration 930: train_error=0.029528, test_error=0.168683
iteration 940: train_error=0.028429, test_error=0.168049
iteration 950: train_error=0.027373, test_error=0.167439
iteration 960: train_error=0.026361, test_error=0.166849
iteration 970: train_error=0.025389, test_error=0.166280
iteration 980: train_error=0.024457, test_error=0.165731
iteration 990: train_error=0.023563, test_error=0.165201
iteration 1000: train_error=0.022706, test_error=0.164689
iteration 1010: train_error=0.021883, test_error=0.164194
iteration 1020: train_error=0.021093, test_error=0.163716
iteration 1030: train_error=0.020335, test_error=0.163255
iteration 1040: train_error=0.019608, test_error=0.162809
iteration 1050: train_error=0.018910, test_error=0.162378
iteration 1060: train_error=0.018240, test_error=0.161961
iteration 1070: train_error=0.017597, test_error=0.161559
iteration 1080: train_error=0.016980, test_error=0.161169
iteration 1090: train_error=0.016387, test_error=0.160793
iteration 1100: train_error=0.015818, test_error=0.160428
iteration 1110: train_error=0.015271, test_error=0.160076
iteration 1120: train_error=0.014746, test_error=0.159735
iteration 1130: train_error=0.014242, test_error=0.159405
iteration 1140: train_error=0.013757, test_error=0.159085
iteration 1150: train_error=0.013291, test_error=0.158776
iteration 1160: train_error=0.012843, test_error=0.158477
iteration 1170: train_error=0.012412, test_error=0.158187
iteration 1180: train_error=0.011998, test_error=0.157906
iteration 1190: train_error=0.011600, test_error=0.157634
iteration 1200: train_error=0.011217, test_error=0.157370
iteration 1210: train_error=0.010849, test_error=0.157114
iteration 1220: train_error=0.010494, test_error=0.156867
iteration 1230: train_error=0.010153, test_error=0.156627
iteration 1240: train_error=0.009825, test_error=0.156394
iteration 1250: train_error=0.009509, test_error=0.156168
iteration 1260: train_error=0.009205, test_error=0.155949
iteration 1270: train_error=0.008912, test_error=0.155737
iteration 1280: train_error=0.008630, test_error=0.155531
iteration 1290: train_error=0.008359, test_error=0.155331
iteration 1300: train_error=0.008097, test_error=0.155136
iteration 1310: train_error=0.007845, test_error=0.154948
iteration 1320: train_error=0.007602, test_error=0.154765
iteration 1330: train_error=0.007368, test_error=0.154587
iteration 1340: train_error=0.007142, test_error=0.154415
iteration 1350: train_error=0.006925, test_error=0.154247
iteration 1360: train_error=0.006715, test_error=0.154084
iteration 1370: train_error=0.006513, test_error=0.153926
iteration 1380: train_error=0.006318, test_error=0.153772
iteration 1390: train_error=0.006130, test_error=0.153623
iteration 1400: train_error=0.005949, test_error=0.153477
iteration 1410: train_error=0.005774, test_error=0.153336
iteration 1420: train_error=0.005605, test_error=0.153198
iteration 1430: train_error=0.005442, test_error=0.153064
iteration 1440: train_error=0.005285, test_error=0.152934
iteration 1450: train_error=0.005133, test_error=0.152808
iteration 1460: train_error=0.004987, test_error=0.152684
iteration 1470: train_error=0.004846, test_error=0.152565
iteration 1480: train_error=0.004709, test_error=0.152448
iteration 1490: train_error=0.004578, test_error=0.152334
iteration 1500: train_error=0.004450, test_error=0.152223
iteration 1510: train_error=0.004328, test_error=0.152115
iteration 1520: train_error=0.004209, test_error=0.152010
iteration 1530: train_error=0.004094, test_error=0.151908
iteration 1540: train_error=0.003984, test_error=0.151808
iteration 1550: train_error=0.003876, test_error=0.151711
iteration 1560: train_error=0.003773, test_error=0.151616
iteration 1570: train_error=0.003673, test_error=0.151524
iteration 1580: train_error=0.003576, test_error=0.151434
iteration 1590: train_error=0.003483, test_error=0.151346
iteration 1600: train_error=0.003393, test_error=0.151260
iteration 1610: train_error=0.003305, test_error=0.151176
iteration 1620: train_error=0.003221, test_error=0.151095
iteration 1630: train_error=0.003139, test_error=0.151015
iteration 1640: train_error=0.003060, test_error=0.150937
iteration 1650: train_error=0.002984, test_error=0.150861
iteration 1660: train_error=0.002910, test_error=0.150787
iteration 1670: train_error=0.002839, test_error=0.150715
iteration 1680: train_error=0.002769, test_error=0.150644
iteration 1690: train_error=0.002702, test_error=0.150575
iteration 1700: train_error=0.002637, test_error=0.150507
iteration 1710: train_error=0.002575, test_error=0.150441
iteration 1720: train_error=0.002514, test_error=0.150377
iteration 1730: train_error=0.002455, test_error=0.150314
iteration 1740: train_error=0.002398, test_error=0.150252
iteration 1750: train_error=0.002343, test_error=0.150192
iteration 1760: train_error=0.002289, test_error=0.150133
iteration 1770: train_error=0.002237, test_error=0.150076
iteration 1780: train_error=0.002187, test_error=0.150019
iteration 1790: train_error=0.002139, test_error=0.149964
iteration 1800: train_error=0.002091, test_error=0.149910
iteration 1810: train_error=0.002046, test_error=0.149858
iteration 1820: train_error=0.002001, test_error=0.149806
iteration 1830: train_error=0.001958, test_error=0.149755
iteration 1840: train_error=0.001917, test_error=0.149706
iteration 1850: train_error=0.001876, test_error=0.149657
iteration 1860: train_error=0.001837, test_error=0.149610
iteration 1870: train_error=0.001799, test_error=0.149563
iteration 1880: train_error=0.001762, test_error=0.149518
iteration 1890: train_error=0.001726, test_error=0.149473
iteration 1900: train_error=0.001692, test_error=0.149429
iteration 1910: train_error=0.001658, test_error=0.149387
iteration 1920: train_error=0.001625, test_error=0.149345
iteration 1930: train_error=0.001593, test_error=0.149303
iteration 1940: train_error=0.001563, test_error=0.149263
iteration 1950: train_error=0.001533, test_error=0.149223
iteration 1960: train_error=0.001503, test_error=0.149184
iteration 1970: train_error=0.001475, test_error=0.149146
iteration 1980: train_error=0.001448, test_error=0.149109
iteration 1990: train_error=0.001421, test_error=0.149072
iteration 2000: train_error=0.001395, test_error=0.149036
[{'train_error': 0.0013949521, 'test_error': 0.14903635}]
Regularized moodel¶
reg_model = build_matrix_norm.build_reg_model(regularization_coeff=0.1, gravity_coeff=1.0, embedding_dim=35,init_stddev=.05)
reg_model.train(num_iterations=2000, learning_rate=20.)
iteration 0: train_error_observed=1.000248, test_error_observed=0.999641, observed_loss=1.000248, regularization_loss=0.017391, gravity_loss=0.000216
iteration 10: train_error_observed=0.998645, test_error_observed=0.999677, observed_loss=0.998645, regularization_loss=0.016990, gravity_loss=0.000206
iteration 20: train_error_observed=0.997083, test_error_observed=0.999689, observed_loss=0.997083, regularization_loss=0.016641, gravity_loss=0.000197
iteration 30: train_error_observed=0.995484, test_error_observed=0.999620, observed_loss=0.995484, regularization_loss=0.016342, gravity_loss=0.000190
iteration 40: train_error_observed=0.993749, test_error_observed=0.999391, observed_loss=0.993749, regularization_loss=0.016092, gravity_loss=0.000184
iteration 50: train_error_observed=0.991718, test_error_observed=0.998866, observed_loss=0.991718, regularization_loss=0.015895, gravity_loss=0.000179
iteration 60: train_error_observed=0.989117, test_error_observed=0.997800, observed_loss=0.989117, regularization_loss=0.015757, gravity_loss=0.000176
iteration 70: train_error_observed=0.985467, test_error_observed=0.995753, observed_loss=0.985467, regularization_loss=0.015693, gravity_loss=0.000175
iteration 80: train_error_observed=0.979937, test_error_observed=0.991946, observed_loss=0.979937, regularization_loss=0.015733, gravity_loss=0.000176
iteration 90: train_error_observed=0.971156, test_error_observed=0.985070, observed_loss=0.971156, regularization_loss=0.015925, gravity_loss=0.000182
iteration 100: train_error_observed=0.957110, test_error_observed=0.973169, observed_loss=0.957110, regularization_loss=0.016351, gravity_loss=0.000197
iteration 110: train_error_observed=0.935512, test_error_observed=0.953945, observed_loss=0.935512, regularization_loss=0.017124, gravity_loss=0.000229
iteration 120: train_error_observed=0.905142, test_error_observed=0.926009, observed_loss=0.905142, regularization_loss=0.018358, gravity_loss=0.000297
iteration 130: train_error_observed=0.867499, test_error_observed=0.890580, observed_loss=0.867499, regularization_loss=0.020104, gravity_loss=0.000425
iteration 140: train_error_observed=0.826201, test_error_observed=0.851164, observed_loss=0.826201, regularization_loss=0.022294, gravity_loss=0.000639
iteration 150: train_error_observed=0.784061, test_error_observed=0.810716, observed_loss=0.784061, regularization_loss=0.024796, gravity_loss=0.000954
iteration 160: train_error_observed=0.742183, test_error_observed=0.770423, observed_loss=0.742183, regularization_loss=0.027501, gravity_loss=0.001381
iteration 170: train_error_observed=0.701360, test_error_observed=0.730989, observed_loss=0.701360, regularization_loss=0.030344, gravity_loss=0.001931
iteration 180: train_error_observed=0.662587, test_error_observed=0.693345, observed_loss=0.662587, regularization_loss=0.033262, gravity_loss=0.002611
iteration 190: train_error_observed=0.626610, test_error_observed=0.658283, observed_loss=0.626610, regularization_loss=0.036185, gravity_loss=0.003417
iteration 200: train_error_observed=0.593694, test_error_observed=0.626165, observed_loss=0.593694, regularization_loss=0.039055, gravity_loss=0.004339
iteration 210: train_error_observed=0.563764, test_error_observed=0.596991, observed_loss=0.563764, regularization_loss=0.041829, gravity_loss=0.005360
iteration 220: train_error_observed=0.536600, test_error_observed=0.570583, observed_loss=0.536600, regularization_loss=0.044485, gravity_loss=0.006464
iteration 230: train_error_observed=0.511940, test_error_observed=0.546701, observed_loss=0.511940, regularization_loss=0.047012, gravity_loss=0.007637
iteration 240: train_error_observed=0.489529, test_error_observed=0.525099, observed_loss=0.489529, regularization_loss=0.049406, gravity_loss=0.008865
iteration 250: train_error_observed=0.469129, test_error_observed=0.505544, observed_loss=0.469129, regularization_loss=0.051669, gravity_loss=0.010136
iteration 260: train_error_observed=0.450522, test_error_observed=0.487819, observed_loss=0.450522, regularization_loss=0.053803, gravity_loss=0.011439
iteration 270: train_error_observed=0.433512, test_error_observed=0.471724, observed_loss=0.433512, regularization_loss=0.055811, gravity_loss=0.012763
iteration 280: train_error_observed=0.417925, test_error_observed=0.457082, observed_loss=0.417925, regularization_loss=0.057699, gravity_loss=0.014099
iteration 290: train_error_observed=0.403605, test_error_observed=0.443731, observed_loss=0.403605, regularization_loss=0.059472, gravity_loss=0.015441
iteration 300: train_error_observed=0.390413, test_error_observed=0.431529, observed_loss=0.390413, regularization_loss=0.061138, gravity_loss=0.016780
iteration 310: train_error_observed=0.378229, test_error_observed=0.420347, observed_loss=0.378229, regularization_loss=0.062701, gravity_loss=0.018111
iteration 320: train_error_observed=0.366946, test_error_observed=0.410077, observed_loss=0.366946, regularization_loss=0.064167, gravity_loss=0.019429
iteration 330: train_error_observed=0.356472, test_error_observed=0.400619, observed_loss=0.356472, regularization_loss=0.065545, gravity_loss=0.020730
iteration 340: train_error_observed=0.346726, test_error_observed=0.391890, observed_loss=0.346726, regularization_loss=0.066838, gravity_loss=0.022011
iteration 350: train_error_observed=0.337636, test_error_observed=0.383815, observed_loss=0.337636, regularization_loss=0.068053, gravity_loss=0.023269
iteration 360: train_error_observed=0.329140, test_error_observed=0.376330, observed_loss=0.329140, regularization_loss=0.069194, gravity_loss=0.024502
iteration 370: train_error_observed=0.321185, test_error_observed=0.369379, observed_loss=0.321185, regularization_loss=0.070268, gravity_loss=0.025707
iteration 380: train_error_observed=0.313721, test_error_observed=0.362912, observed_loss=0.313721, regularization_loss=0.071279, gravity_loss=0.026883
iteration 390: train_error_observed=0.306707, test_error_observed=0.356885, observed_loss=0.306707, regularization_loss=0.072231, gravity_loss=0.028030
iteration 400: train_error_observed=0.300105, test_error_observed=0.351261, observed_loss=0.300105, regularization_loss=0.073127, gravity_loss=0.029146
iteration 410: train_error_observed=0.293881, test_error_observed=0.346004, observed_loss=0.293881, regularization_loss=0.073973, gravity_loss=0.030230
iteration 420: train_error_observed=0.288005, test_error_observed=0.341084, observed_loss=0.288005, regularization_loss=0.074771, gravity_loss=0.031282
iteration 430: train_error_observed=0.282450, test_error_observed=0.336474, observed_loss=0.282450, regularization_loss=0.075525, gravity_loss=0.032301
iteration 440: train_error_observed=0.277191, test_error_observed=0.332150, observed_loss=0.277191, regularization_loss=0.076237, gravity_loss=0.033288
iteration 450: train_error_observed=0.272207, test_error_observed=0.328089, observed_loss=0.272207, regularization_loss=0.076911, gravity_loss=0.034242
iteration 460: train_error_observed=0.267478, test_error_observed=0.324271, observed_loss=0.267478, regularization_loss=0.077550, gravity_loss=0.035164
iteration 470: train_error_observed=0.262984, test_error_observed=0.320679, observed_loss=0.262984, regularization_loss=0.078154, gravity_loss=0.036052
iteration 480: train_error_observed=0.258710, test_error_observed=0.317295, observed_loss=0.258710, regularization_loss=0.078728, gravity_loss=0.036909
iteration 490: train_error_observed=0.254639, test_error_observed=0.314104, observed_loss=0.254639, regularization_loss=0.079273, gravity_loss=0.037734
iteration 500: train_error_observed=0.250757, test_error_observed=0.311093, observed_loss=0.250757, regularization_loss=0.079791, gravity_loss=0.038528
iteration 510: train_error_observed=0.247052, test_error_observed=0.308249, observed_loss=0.247052, regularization_loss=0.080285, gravity_loss=0.039291
iteration 520: train_error_observed=0.243511, test_error_observed=0.305560, observed_loss=0.243511, regularization_loss=0.080755, gravity_loss=0.040023
iteration 530: train_error_observed=0.240123, test_error_observed=0.303015, observed_loss=0.240123, regularization_loss=0.081204, gravity_loss=0.040726
iteration 540: train_error_observed=0.236877, test_error_observed=0.300605, observed_loss=0.236877, regularization_loss=0.081634, gravity_loss=0.041400
iteration 550: train_error_observed=0.233765, test_error_observed=0.298320, observed_loss=0.233765, regularization_loss=0.082045, gravity_loss=0.042046
iteration 560: train_error_observed=0.230777, test_error_observed=0.296153, observed_loss=0.230777, regularization_loss=0.082440, gravity_loss=0.042664
iteration 570: train_error_observed=0.227905, test_error_observed=0.294095, observed_loss=0.227905, regularization_loss=0.082820, gravity_loss=0.043256
iteration 580: train_error_observed=0.225142, test_error_observed=0.292139, observed_loss=0.225142, regularization_loss=0.083185, gravity_loss=0.043822
iteration 590: train_error_observed=0.222480, test_error_observed=0.290278, observed_loss=0.222480, regularization_loss=0.083538, gravity_loss=0.044362
iteration 600: train_error_observed=0.219914, test_error_observed=0.288507, observed_loss=0.219914, regularization_loss=0.083879, gravity_loss=0.044879
iteration 610: train_error_observed=0.217437, test_error_observed=0.286820, observed_loss=0.217437, regularization_loss=0.084209, gravity_loss=0.045372
iteration 620: train_error_observed=0.215043, test_error_observed=0.285211, observed_loss=0.215043, regularization_loss=0.084530, gravity_loss=0.045842
iteration 630: train_error_observed=0.212727, test_error_observed=0.283676, observed_loss=0.212727, regularization_loss=0.084842, gravity_loss=0.046290
iteration 640: train_error_observed=0.210484, test_error_observed=0.282210, observed_loss=0.210484, regularization_loss=0.085145, gravity_loss=0.046716
iteration 650: train_error_observed=0.208311, test_error_observed=0.280809, observed_loss=0.208311, regularization_loss=0.085442, gravity_loss=0.047123
iteration 660: train_error_observed=0.206202, test_error_observed=0.279469, observed_loss=0.206202, regularization_loss=0.085733, gravity_loss=0.047509
iteration 670: train_error_observed=0.204154, test_error_observed=0.278186, observed_loss=0.204154, regularization_loss=0.086017, gravity_loss=0.047877
iteration 680: train_error_observed=0.202164, test_error_observed=0.276958, observed_loss=0.202164, regularization_loss=0.086297, gravity_loss=0.048226
iteration 690: train_error_observed=0.200227, test_error_observed=0.275780, observed_loss=0.200227, regularization_loss=0.086572, gravity_loss=0.048558
iteration 700: train_error_observed=0.198341, test_error_observed=0.274650, observed_loss=0.198341, regularization_loss=0.086843, gravity_loss=0.048873
iteration 710: train_error_observed=0.196504, test_error_observed=0.273566, observed_loss=0.196504, regularization_loss=0.087111, gravity_loss=0.049171
iteration 720: train_error_observed=0.194711, test_error_observed=0.272525, observed_loss=0.194711, regularization_loss=0.087376, gravity_loss=0.049454
iteration 730: train_error_observed=0.192961, test_error_observed=0.271524, observed_loss=0.192961, regularization_loss=0.087638, gravity_loss=0.049721
iteration 740: train_error_observed=0.191251, test_error_observed=0.270561, observed_loss=0.191251, regularization_loss=0.087899, gravity_loss=0.049974
iteration 750: train_error_observed=0.189579, test_error_observed=0.269635, observed_loss=0.189579, regularization_loss=0.088157, gravity_loss=0.050213
iteration 760: train_error_observed=0.187944, test_error_observed=0.268743, observed_loss=0.187944, regularization_loss=0.088414, gravity_loss=0.050439
iteration 770: train_error_observed=0.186343, test_error_observed=0.267883, observed_loss=0.186343, regularization_loss=0.088670, gravity_loss=0.050652
iteration 780: train_error_observed=0.184774, test_error_observed=0.267054, observed_loss=0.184774, regularization_loss=0.088924, gravity_loss=0.050852
iteration 790: train_error_observed=0.183236, test_error_observed=0.266255, observed_loss=0.183236, regularization_loss=0.089179, gravity_loss=0.051041
iteration 800: train_error_observed=0.181727, test_error_observed=0.265483, observed_loss=0.181727, regularization_loss=0.089432, gravity_loss=0.051218
iteration 810: train_error_observed=0.180246, test_error_observed=0.264738, observed_loss=0.180246, regularization_loss=0.089686, gravity_loss=0.051384
iteration 820: train_error_observed=0.178792, test_error_observed=0.264018, observed_loss=0.178792, regularization_loss=0.089939, gravity_loss=0.051540
iteration 830: train_error_observed=0.177363, test_error_observed=0.263322, observed_loss=0.177363, regularization_loss=0.090193, gravity_loss=0.051685
iteration 840: train_error_observed=0.175959, test_error_observed=0.262649, observed_loss=0.175959, regularization_loss=0.090447, gravity_loss=0.051821
iteration 850: train_error_observed=0.174577, test_error_observed=0.261998, observed_loss=0.174577, regularization_loss=0.090701, gravity_loss=0.051947
iteration 860: train_error_observed=0.173218, test_error_observed=0.261368, observed_loss=0.173218, regularization_loss=0.090956, gravity_loss=0.052065
iteration 870: train_error_observed=0.171880, test_error_observed=0.260757, observed_loss=0.171880, regularization_loss=0.091211, gravity_loss=0.052174
iteration 880: train_error_observed=0.170562, test_error_observed=0.260165, observed_loss=0.170562, regularization_loss=0.091466, gravity_loss=0.052275
iteration 890: train_error_observed=0.169264, test_error_observed=0.259592, observed_loss=0.169264, regularization_loss=0.091723, gravity_loss=0.052367
iteration 900: train_error_observed=0.167985, test_error_observed=0.259036, observed_loss=0.167985, regularization_loss=0.091980, gravity_loss=0.052452
iteration 910: train_error_observed=0.166724, test_error_observed=0.258496, observed_loss=0.166724, regularization_loss=0.092238, gravity_loss=0.052530
iteration 920: train_error_observed=0.165480, test_error_observed=0.257973, observed_loss=0.165480, regularization_loss=0.092497, gravity_loss=0.052601
iteration 930: train_error_observed=0.164254, test_error_observed=0.257465, observed_loss=0.164254, regularization_loss=0.092756, gravity_loss=0.052665
iteration 940: train_error_observed=0.163044, test_error_observed=0.256971, observed_loss=0.163044, regularization_loss=0.093016, gravity_loss=0.052723
iteration 950: train_error_observed=0.161850, test_error_observed=0.256492, observed_loss=0.161850, regularization_loss=0.093277, gravity_loss=0.052775
iteration 960: train_error_observed=0.160671, test_error_observed=0.256026, observed_loss=0.160671, regularization_loss=0.093539, gravity_loss=0.052820
iteration 970: train_error_observed=0.159508, test_error_observed=0.255574, observed_loss=0.159508, regularization_loss=0.093802, gravity_loss=0.052860
iteration 980: train_error_observed=0.158359, test_error_observed=0.255134, observed_loss=0.158359, regularization_loss=0.094065, gravity_loss=0.052895
iteration 990: train_error_observed=0.157225, test_error_observed=0.254706, observed_loss=0.157225, regularization_loss=0.094329, gravity_loss=0.052924
iteration 1000: train_error_observed=0.156104, test_error_observed=0.254289, observed_loss=0.156104, regularization_loss=0.094593, gravity_loss=0.052948
iteration 1010: train_error_observed=0.154998, test_error_observed=0.253884, observed_loss=0.154998, regularization_loss=0.094859, gravity_loss=0.052968
iteration 1020: train_error_observed=0.153904, test_error_observed=0.253490, observed_loss=0.153904, regularization_loss=0.095124, gravity_loss=0.052983
iteration 1030: train_error_observed=0.152824, test_error_observed=0.253106, observed_loss=0.152824, regularization_loss=0.095391, gravity_loss=0.052994
iteration 1040: train_error_observed=0.151757, test_error_observed=0.252732, observed_loss=0.151757, regularization_loss=0.095657, gravity_loss=0.053000
iteration 1050: train_error_observed=0.150702, test_error_observed=0.252368, observed_loss=0.150702, regularization_loss=0.095924, gravity_loss=0.053002
iteration 1060: train_error_observed=0.149660, test_error_observed=0.252014, observed_loss=0.149660, regularization_loss=0.096192, gravity_loss=0.053001
iteration 1070: train_error_observed=0.148630, test_error_observed=0.251668, observed_loss=0.148630, regularization_loss=0.096459, gravity_loss=0.052996
iteration 1080: train_error_observed=0.147613, test_error_observed=0.251332, observed_loss=0.147613, regularization_loss=0.096727, gravity_loss=0.052987
iteration 1090: train_error_observed=0.146607, test_error_observed=0.251004, observed_loss=0.146607, regularization_loss=0.096995, gravity_loss=0.052975
iteration 1100: train_error_observed=0.145613, test_error_observed=0.250684, observed_loss=0.145613, regularization_loss=0.097263, gravity_loss=0.052960
iteration 1110: train_error_observed=0.144631, test_error_observed=0.250372, observed_loss=0.144631, regularization_loss=0.097531, gravity_loss=0.052941
iteration 1120: train_error_observed=0.143660, test_error_observed=0.250068, observed_loss=0.143660, regularization_loss=0.097799, gravity_loss=0.052920
iteration 1130: train_error_observed=0.142700, test_error_observed=0.249772, observed_loss=0.142700, regularization_loss=0.098066, gravity_loss=0.052896
iteration 1140: train_error_observed=0.141752, test_error_observed=0.249483, observed_loss=0.141752, regularization_loss=0.098333, gravity_loss=0.052869
iteration 1150: train_error_observed=0.140815, test_error_observed=0.249201, observed_loss=0.140815, regularization_loss=0.098600, gravity_loss=0.052840
iteration 1160: train_error_observed=0.139889, test_error_observed=0.248926, observed_loss=0.139889, regularization_loss=0.098867, gravity_loss=0.052808
iteration 1170: train_error_observed=0.138973, test_error_observed=0.248657, observed_loss=0.138973, regularization_loss=0.099133, gravity_loss=0.052774
iteration 1180: train_error_observed=0.138069, test_error_observed=0.248395, observed_loss=0.138069, regularization_loss=0.099398, gravity_loss=0.052738
iteration 1190: train_error_observed=0.137175, test_error_observed=0.248140, observed_loss=0.137175, regularization_loss=0.099663, gravity_loss=0.052700
iteration 1200: train_error_observed=0.136292, test_error_observed=0.247890, observed_loss=0.136292, regularization_loss=0.099927, gravity_loss=0.052660
iteration 1210: train_error_observed=0.135420, test_error_observed=0.247647, observed_loss=0.135420, regularization_loss=0.100190, gravity_loss=0.052618
iteration 1220: train_error_observed=0.134558, test_error_observed=0.247410, observed_loss=0.134558, regularization_loss=0.100453, gravity_loss=0.052574
iteration 1230: train_error_observed=0.133706, test_error_observed=0.247178, observed_loss=0.133706, regularization_loss=0.100714, gravity_loss=0.052528
iteration 1240: train_error_observed=0.132864, test_error_observed=0.246951, observed_loss=0.132864, regularization_loss=0.100975, gravity_loss=0.052481
iteration 1250: train_error_observed=0.132033, test_error_observed=0.246730, observed_loss=0.132033, regularization_loss=0.101234, gravity_loss=0.052432
iteration 1260: train_error_observed=0.131211, test_error_observed=0.246514, observed_loss=0.131211, regularization_loss=0.101492, gravity_loss=0.052382
iteration 1270: train_error_observed=0.130400, test_error_observed=0.246304, observed_loss=0.130400, regularization_loss=0.101749, gravity_loss=0.052331
iteration 1280: train_error_observed=0.129598, test_error_observed=0.246098, observed_loss=0.129598, regularization_loss=0.102005, gravity_loss=0.052278
iteration 1290: train_error_observed=0.128806, test_error_observed=0.245897, observed_loss=0.128806, regularization_loss=0.102260, gravity_loss=0.052224
iteration 1300: train_error_observed=0.128024, test_error_observed=0.245701, observed_loss=0.128024, regularization_loss=0.102513, gravity_loss=0.052169
iteration 1310: train_error_observed=0.127251, test_error_observed=0.245509, observed_loss=0.127251, regularization_loss=0.102765, gravity_loss=0.052112
iteration 1320: train_error_observed=0.126488, test_error_observed=0.245322, observed_loss=0.126488, regularization_loss=0.103016, gravity_loss=0.052055
iteration 1330: train_error_observed=0.125735, test_error_observed=0.245139, observed_loss=0.125735, regularization_loss=0.103265, gravity_loss=0.051997
iteration 1340: train_error_observed=0.124990, test_error_observed=0.244961, observed_loss=0.124990, regularization_loss=0.103512, gravity_loss=0.051938
iteration 1350: train_error_observed=0.124255, test_error_observed=0.244786, observed_loss=0.124255, regularization_loss=0.103758, gravity_loss=0.051878
iteration 1360: train_error_observed=0.123528, test_error_observed=0.244616, observed_loss=0.123528, regularization_loss=0.104002, gravity_loss=0.051817
iteration 1370: train_error_observed=0.122811, test_error_observed=0.244450, observed_loss=0.122811, regularization_loss=0.104245, gravity_loss=0.051756
iteration 1380: train_error_observed=0.122103, test_error_observed=0.244287, observed_loss=0.122103, regularization_loss=0.104486, gravity_loss=0.051694
iteration 1390: train_error_observed=0.121403, test_error_observed=0.244128, observed_loss=0.121403, regularization_loss=0.104726, gravity_loss=0.051631
iteration 1400: train_error_observed=0.120712, test_error_observed=0.243973, observed_loss=0.120712, regularization_loss=0.104963, gravity_loss=0.051568
iteration 1410: train_error_observed=0.120030, test_error_observed=0.243822, observed_loss=0.120030, regularization_loss=0.105199, gravity_loss=0.051504
iteration 1420: train_error_observed=0.119356, test_error_observed=0.243674, observed_loss=0.119356, regularization_loss=0.105434, gravity_loss=0.051440
iteration 1430: train_error_observed=0.118691, test_error_observed=0.243529, observed_loss=0.118691, regularization_loss=0.105666, gravity_loss=0.051375
iteration 1440: train_error_observed=0.118033, test_error_observed=0.243388, observed_loss=0.118033, regularization_loss=0.105897, gravity_loss=0.051310
iteration 1450: train_error_observed=0.117384, test_error_observed=0.243250, observed_loss=0.117384, regularization_loss=0.106126, gravity_loss=0.051245
iteration 1460: train_error_observed=0.116743, test_error_observed=0.243115, observed_loss=0.116743, regularization_loss=0.106353, gravity_loss=0.051179
iteration 1470: train_error_observed=0.116110, test_error_observed=0.242983, observed_loss=0.116110, regularization_loss=0.106578, gravity_loss=0.051113
iteration 1480: train_error_observed=0.115485, test_error_observed=0.242854, observed_loss=0.115485, regularization_loss=0.106801, gravity_loss=0.051047
iteration 1490: train_error_observed=0.114867, test_error_observed=0.242728, observed_loss=0.114867, regularization_loss=0.107023, gravity_loss=0.050980
iteration 1500: train_error_observed=0.114257, test_error_observed=0.242606, observed_loss=0.114257, regularization_loss=0.107243, gravity_loss=0.050913
iteration 1510: train_error_observed=0.113655, test_error_observed=0.242485, observed_loss=0.113655, regularization_loss=0.107461, gravity_loss=0.050847
iteration 1520: train_error_observed=0.113060, test_error_observed=0.242368, observed_loss=0.113060, regularization_loss=0.107677, gravity_loss=0.050779
iteration 1530: train_error_observed=0.112472, test_error_observed=0.242253, observed_loss=0.112472, regularization_loss=0.107891, gravity_loss=0.050712
iteration 1540: train_error_observed=0.111892, test_error_observed=0.242141, observed_loss=0.111892, regularization_loss=0.108103, gravity_loss=0.050645
iteration 1550: train_error_observed=0.111318, test_error_observed=0.242032, observed_loss=0.111318, regularization_loss=0.108314, gravity_loss=0.050578
iteration 1560: train_error_observed=0.110752, test_error_observed=0.241925, observed_loss=0.110752, regularization_loss=0.108523, gravity_loss=0.050510
iteration 1570: train_error_observed=0.110192, test_error_observed=0.241821, observed_loss=0.110192, regularization_loss=0.108729, gravity_loss=0.050443
iteration 1580: train_error_observed=0.109640, test_error_observed=0.241719, observed_loss=0.109640, regularization_loss=0.108934, gravity_loss=0.050376
iteration 1590: train_error_observed=0.109094, test_error_observed=0.241619, observed_loss=0.109094, regularization_loss=0.109138, gravity_loss=0.050308
iteration 1600: train_error_observed=0.108555, test_error_observed=0.241521, observed_loss=0.108555, regularization_loss=0.109339, gravity_loss=0.050241
iteration 1610: train_error_observed=0.108022, test_error_observed=0.241426, observed_loss=0.108022, regularization_loss=0.109539, gravity_loss=0.050174
iteration 1620: train_error_observed=0.107495, test_error_observed=0.241333, observed_loss=0.107495, regularization_loss=0.109737, gravity_loss=0.050106
iteration 1630: train_error_observed=0.106975, test_error_observed=0.241242, observed_loss=0.106975, regularization_loss=0.109933, gravity_loss=0.050039
iteration 1640: train_error_observed=0.106462, test_error_observed=0.241154, observed_loss=0.106462, regularization_loss=0.110127, gravity_loss=0.049972
iteration 1650: train_error_observed=0.105954, test_error_observed=0.241067, observed_loss=0.105954, regularization_loss=0.110320, gravity_loss=0.049905
iteration 1660: train_error_observed=0.105452, test_error_observed=0.240982, observed_loss=0.105452, regularization_loss=0.110510, gravity_loss=0.049838
iteration 1670: train_error_observed=0.104957, test_error_observed=0.240899, observed_loss=0.104957, regularization_loss=0.110700, gravity_loss=0.049772
iteration 1680: train_error_observed=0.104467, test_error_observed=0.240819, observed_loss=0.104467, regularization_loss=0.110887, gravity_loss=0.049705
iteration 1690: train_error_observed=0.103983, test_error_observed=0.240740, observed_loss=0.103983, regularization_loss=0.111073, gravity_loss=0.049639
iteration 1700: train_error_observed=0.103505, test_error_observed=0.240662, observed_loss=0.103505, regularization_loss=0.111257, gravity_loss=0.049573
iteration 1710: train_error_observed=0.103032, test_error_observed=0.240587, observed_loss=0.103032, regularization_loss=0.111439, gravity_loss=0.049507
iteration 1720: train_error_observed=0.102565, test_error_observed=0.240514, observed_loss=0.102565, regularization_loss=0.111620, gravity_loss=0.049441
iteration 1730: train_error_observed=0.102103, test_error_observed=0.240442, observed_loss=0.102103, regularization_loss=0.111799, gravity_loss=0.049375
iteration 1740: train_error_observed=0.101647, test_error_observed=0.240372, observed_loss=0.101647, regularization_loss=0.111976, gravity_loss=0.049310
iteration 1750: train_error_observed=0.101196, test_error_observed=0.240303, observed_loss=0.101196, regularization_loss=0.112152, gravity_loss=0.049244
iteration 1760: train_error_observed=0.100750, test_error_observed=0.240236, observed_loss=0.100750, regularization_loss=0.112327, gravity_loss=0.049179
iteration 1770: train_error_observed=0.100310, test_error_observed=0.240171, observed_loss=0.100310, regularization_loss=0.112499, gravity_loss=0.049115
iteration 1780: train_error_observed=0.099874, test_error_observed=0.240107, observed_loss=0.099874, regularization_loss=0.112670, gravity_loss=0.049050
iteration 1790: train_error_observed=0.099443, test_error_observed=0.240045, observed_loss=0.099443, regularization_loss=0.112840, gravity_loss=0.048986
iteration 1800: train_error_observed=0.099018, test_error_observed=0.239984, observed_loss=0.099018, regularization_loss=0.113008, gravity_loss=0.048922
iteration 1810: train_error_observed=0.098597, test_error_observed=0.239925, observed_loss=0.098597, regularization_loss=0.113175, gravity_loss=0.048858
iteration 1820: train_error_observed=0.098180, test_error_observed=0.239867, observed_loss=0.098180, regularization_loss=0.113340, gravity_loss=0.048794
iteration 1830: train_error_observed=0.097769, test_error_observed=0.239810, observed_loss=0.097769, regularization_loss=0.113504, gravity_loss=0.048731
iteration 1840: train_error_observed=0.097362, test_error_observed=0.239755, observed_loss=0.097362, regularization_loss=0.113666, gravity_loss=0.048668
iteration 1850: train_error_observed=0.096960, test_error_observed=0.239701, observed_loss=0.096960, regularization_loss=0.113827, gravity_loss=0.048605
iteration 1860: train_error_observed=0.096562, test_error_observed=0.239648, observed_loss=0.096562, regularization_loss=0.113986, gravity_loss=0.048543
iteration 1870: train_error_observed=0.096169, test_error_observed=0.239597, observed_loss=0.096169, regularization_loss=0.114144, gravity_loss=0.048481
iteration 1880: train_error_observed=0.095780, test_error_observed=0.239547, observed_loss=0.095780, regularization_loss=0.114301, gravity_loss=0.048419
iteration 1890: train_error_observed=0.095395, test_error_observed=0.239498, observed_loss=0.095395, regularization_loss=0.114456, gravity_loss=0.048357
iteration 1900: train_error_observed=0.095014, test_error_observed=0.239451, observed_loss=0.095014, regularization_loss=0.114610, gravity_loss=0.048296
iteration 1910: train_error_observed=0.094638, test_error_observed=0.239404, observed_loss=0.094638, regularization_loss=0.114763, gravity_loss=0.048234
iteration 1920: train_error_observed=0.094266, test_error_observed=0.239359, observed_loss=0.094266, regularization_loss=0.114914, gravity_loss=0.048174
iteration 1930: train_error_observed=0.093897, test_error_observed=0.239315, observed_loss=0.093897, regularization_loss=0.115064, gravity_loss=0.048113
iteration 1940: train_error_observed=0.093533, test_error_observed=0.239272, observed_loss=0.093533, regularization_loss=0.115213, gravity_loss=0.048053
iteration 1950: train_error_observed=0.093173, test_error_observed=0.239230, observed_loss=0.093173, regularization_loss=0.115360, gravity_loss=0.047993
iteration 1960: train_error_observed=0.092816, test_error_observed=0.239189, observed_loss=0.092816, regularization_loss=0.115506, gravity_loss=0.047933
iteration 1970: train_error_observed=0.092464, test_error_observed=0.239150, observed_loss=0.092464, regularization_loss=0.115651, gravity_loss=0.047874
iteration 1980: train_error_observed=0.092115, test_error_observed=0.239111, observed_loss=0.092115, regularization_loss=0.115795, gravity_loss=0.047814
iteration 1990: train_error_observed=0.091770, test_error_observed=0.239073, observed_loss=0.091770, regularization_loss=0.115937, gravity_loss=0.047756
iteration 2000: train_error_observed=0.091428, test_error_observed=0.239036, observed_loss=0.091428, regularization_loss=0.116079, gravity_loss=0.047697
[{'train_error_observed': 0.09142829, 'test_error_observed': 0.23903622},
{'observed_loss': 0.09142829,
'regularization_loss': 0.11607879,
'gravity_loss': 0.04769704}]
In both models, we observe a steep loss in train error and test as the model progress. Although, the regularized model has a higher MSE, both on the training and test set. It must be noted that the quality of recommendation is improved when regularization is added, which is proven when the artist_neighbors() function is utilized. In addition, we observe in the end evaluation section, that the the performance of the model is improved when regularization is added. The test error decreases similarity to the test error, although it plateaus around the 1000 epoch mark. As expected, the the additional loss generated by the regularization functions increases over epochs. We add the following regularisation terms to our model.
Regularization of the model parameters. This is a common \(\ell_2\) regularization term on the embedding matrices, given by \(r(U, V) = \frac{1}{N} \sum_i \|U_i\|^2 + \frac{1}{M}\sum_j \|V_j\|^2\).
A global prior that pushes the prediction of any pair towards zero, called the gravity term. This is given by \(g(U, V) = \frac{1}{MN} \sum_{i = 1}^N \sum_{j = 1}^M \langle U_i, V_j \rangle^2\)
These terms modifies the “global” loss (as in, the sum of the network loss and the regularization loss) in order to drive the optimization algorithm in desired directions i.e. prevent overfitting.
Evaluating the embeddings¶
We will use two similairty meausres to inspect the robustness of our system:
Dot product: score of artist j \(\langle u, V_j \rangle\).
Cosine angle: score of artist j \(\frac{\langle u, V_j \rangle}{\|u\|\|V_j\|}\).
DOT = 'dot'
COSINE = 'cosine'
def compute_scores(query_embedding, item_embeddings, measure=DOT):
"""Computes the scores of the candidates given a query.
Args:
query_embedding: a vector of shape [k], representing the query embedding.
item_embeddings: a matrix of shape [N, k], such that row i is the embedding
of item i.
measure: a string specifying the similarity measure to be used. Can be
either DOT or COSINE.
Returns:
scores: a vector of shape [N], such that scores[i] is the score of item i.
"""
u = query_embedding
V = item_embeddings
if measure == COSINE:
V = V / np.linalg.norm(V, axis=1, keepdims=True)
u = u / np.linalg.norm(u)
scores = u.dot(V.T)
return scores
def user_recommendations(model,user_id, k=15, measure=DOT, exclude_rated=False):
scores = compute_scores(
model.embeddings["userID"][user_id], model.embeddings["artistID"], measure)
score_key = measure + ' score'
df = pd.DataFrame({
'score': list(scores),
'name': artists.sort_values('artistID', ascending=True)['name'],
'most assigned tag':artists.sort_values('artistID', ascending=True)['mostCommonGenre']
})
return df.sort_values(['score'], ascending=False).head(k)
def artist_neighbors(model, title_substring, measure=DOT, k=6):
# Search for artist ids that match the given substring.
inv_artist_id_mapping = {v: k for k, v in orginal_artist_ids.items()}
ids = artists[artists['name'].str.contains(title_substring)].artistID.values
titles = artists[artists.artistID.isin(ids)]['name'].values
if len(titles) == 0:
raise ValueError("Found no artists with name %s" % title_substring)
print("Nearest neighbors of : %s." % titles[0])
if len(titles) > 1:
print("[Found more than one matching artist. Other candidates: {}]".format(
", ".join(titles[1:])))
artists_id_orginal = ids[0]
asrtists_id_mapped = inv_artist_id_mapping[ids[0]]
scores = compute_scores(
model.embeddings["artistID"][asrtists_id_mapped], model.embeddings["artistID"],
measure)
score_key = measure + ' score'
df = pd.DataFrame({
score_key: list(scores),
'name': artists.sort_values('artistID', ascending=True)['name'],
'most assigned tag':artists.sort_values('artistID', ascending=True)['mostCommonGenre']
})
return df.sort_values([score_key], ascending=False).head(k)
Here, we find the most similar artists to the band the cure. We also include the most assigned tag associated with an artist. The reccomdations are conistent with our domain knowedge of bands similar to the cure.
artist_neighbors(vanilla_model, "The Cure", DOT)
Nearest neighbors of : The Cure.
| dot score | name | most assigned tag | |
|---|---|---|---|
| 9437 | 0.550 | The Cure | chillout |
| 12363 | 0.543 | Muse | chillout |
| 16680 | 0.542 | The Beatles | chillout |
| 14252 | 0.541 | Damien Rice | chillout |
| 17278 | 0.540 | Kings of Leon | chillout |
| 48852 | 0.539 | Legião Urbana | 80s |
artist_neighbors(vanilla_model, "The Cure", COSINE)
Nearest neighbors of : The Cure.
| cosine score | name | most assigned tag | |
|---|---|---|---|
| 9437 | 1.000 | The Cure | chillout |
| 8273 | 0.976 | Radiohead | chillout |
| 16680 | 0.971 | The Beatles | chillout |
| 10850 | 0.968 | Placebo | chillout |
| 4936 | 0.964 | Depeche Mode | chillout |
| 14553 | 0.964 | Arctic Monkeys | chillout |
artist_neighbors(reg_model, "The Cure", DOT)
Nearest neighbors of : The Cure.
| dot score | name | most assigned tag | |
|---|---|---|---|
| 9437 | 3.301 | The Cure | chillout |
| 16680 | 3.265 | The Beatles | chillout |
| 18364 | 3.251 | Nirvana | pop |
| 15847 | 3.231 | Red Hot Chili Peppers | chillout |
| 12363 | 3.228 | Muse | chillout |
| 17832 | 3.197 | Green Day | chillout |
artist_neighbors(reg_model, "The Cure", COSINE)
Nearest neighbors of : The Cure.
| cosine score | name | most assigned tag | |
|---|---|---|---|
| 9437 | 1.000 | The Cure | chillout |
| 38968 | 0.961 | U2 | electronic |
| 32942 | 0.960 | The Smiths | groove |
| 4936 | 0.954 | Depeche Mode | chillout |
| 33564 | 0.952 | The Smashing Pumpkins | atmospheric |
| 15847 | 0.947 | Red Hot Chili Peppers | chillout |
We observe that dot product tends to recommends more popular artists such as Nirvana and The Beatles, where as Cosine Similarity recommends more obscure artists. This is likely due to the fact that the norm of the embedding in matrix factorization is often correlated with prevalence. The regularised model seems to output better reccomodations as the varation of the most assigned tag attribute is less when compared to the vanilla model. In addition, Marilyn Manson was recommended by the vanilla model in our intial run. We argue that these artists are most dis-similar! However, this observation is subject to change when you run the model, as we initialize the embedddings with a random gaussian generator.
def artist_embedding_norm(models):
"""Visualizes the norm and number of ratings of the artist embeddings.
Args:
model: A train_matrix_norm object.
"""
if not isinstance(models, list):
models = [models]
df = pd.DataFrame({
'name': artists.sort_values('artistID', ascending=True)['name'].values,
'number of user-artist interactions': user_artists[['artistID','userID']].sort_values('artistID', ascending=True).groupby('artistID').count()['userID'].values,
})
charts = []
brush = alt.selection_interval()
for i, model in enumerate(models):
norm_key = 'norm'+str(i)
df[norm_key] = np.linalg.norm(model.embeddings["artistID"], axis=1)
nearest = alt.selection(
type='single', encodings=['x', 'y'], on='mouseover', nearest=True,
empty='none')
base = alt.Chart().mark_circle().encode(
x='number of user-artist interactions',
y=norm_key,
color=alt.condition(brush, alt.value('#4c78a8'), alt.value('lightgray'))
).properties(
selection=nearest).add_selection(brush)
text = alt.Chart().mark_text(align='center', dx=5, dy=-5).encode(
x='number of user-artist interactions', y=norm_key,
text=alt.condition(nearest, 'name', alt.value('')))
charts.append(alt.layer(base, text))
return alt.hconcat(*charts, data=df)
artist_embedding_norm(reg_model)
def visualize_movie_embeddings(data, x, y):
genre_filter = alt.selection_multi(fields=['top10TagValue'])
genre_chart = alt.Chart().mark_bar().encode(
x="count()",
y=alt.Y('top10TagValue'),
color=alt.condition(
genre_filter,
alt.Color("top10TagValue:N"),
alt.value('lightgray'))
).properties(height=300, selection=genre_filter)
nearest = alt.selection(
type='single', encodings=['x', 'y'], on='mouseover', nearest=True,
empty='none')
base = alt.Chart().mark_circle().encode(
x=x,
y=y,
color=alt.condition(genre_filter, "top10TagValue", alt.value("whitesmoke")),
).properties(
width=600,
height=600,
selection=nearest)
text = alt.Chart().mark_text(align='left', dx=5, dy=-5).encode(
x=x,
y=y,
text=alt.condition(nearest, 'name', alt.value('')))
return alt.hconcat(alt.layer(base, text), genre_chart, data=data)
def tsne_movie_embeddings(model):
"""Visualizes the movie embeddings, projected using t-SNE with Cosine measure.
Args:
model: A MFModel object.
"""
tsne = sklearn.manifold.TSNE(
n_components=2, perplexity=40, metric='cosine', early_exaggeration=10.0,
init='pca', verbose=True, n_iter=400)
print('Running t-SNE...')
V_proj = tsne.fit_transform(model.embeddings["artistID"])
artists.loc[:,'x'] = V_proj[:, 0]
artists.loc[:,'y'] = V_proj[:, 1]
return visualize_movie_embeddings(artists, 'x', 'y')
T-distributed stochastic neighbor embedding (t-SNE) is a dimensionality reduction algorithm useful for visualizing high dimensional data. We use this algorithim to visualise our embeddings of the regualrised model. Due to the large number of user submitted semantic categories, we decide to color-code the top 15 tags, with the rest being labelled as ‘N/A’. Although the sea of orange, indicating’N/A’, makes it difficult to interrupt these results, the regularised model seems to adequaltly cluster artists of a similar genre in it’s embeddings.
tsne_movie_embeddings(reg_model)
Running t-SNE...
[t-SNE] Computing 121 nearest neighbors...
[t-SNE] Indexed 17632 samples in 0.001s...
/opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/sklearn/manifold/_t_sne.py:793: FutureWarning: The default learning rate in TSNE will change from 200.0 to 'auto' in 1.2.
FutureWarning,
/opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/sklearn/manifold/_t_sne.py:827: FutureWarning: 'square_distances' has been introduced in 0.24 to help phase out legacy squaring behavior. The 'legacy' setting will be removed in 1.1 (renaming of 0.26), and the default setting will be changed to True. In 1.3, 'square_distances' will be removed altogether, and distances will be squared by default. Set 'square_distances'=True to silence this warning.
FutureWarning,
[t-SNE] Computed neighbors for 17632 samples in 4.598s...
[t-SNE] Computed conditional probabilities for sample 1000 / 17632
[t-SNE] Computed conditional probabilities for sample 2000 / 17632
[t-SNE] Computed conditional probabilities for sample 3000 / 17632
[t-SNE] Computed conditional probabilities for sample 4000 / 17632
[t-SNE] Computed conditional probabilities for sample 5000 / 17632
[t-SNE] Computed conditional probabilities for sample 6000 / 17632
[t-SNE] Computed conditional probabilities for sample 7000 / 17632
[t-SNE] Computed conditional probabilities for sample 8000 / 17632
[t-SNE] Computed conditional probabilities for sample 9000 / 17632
[t-SNE] Computed conditional probabilities for sample 10000 / 17632
[t-SNE] Computed conditional probabilities for sample 11000 / 17632
[t-SNE] Computed conditional probabilities for sample 12000 / 17632
[t-SNE] Computed conditional probabilities for sample 13000 / 17632
[t-SNE] Computed conditional probabilities for sample 14000 / 17632
[t-SNE] Computed conditional probabilities for sample 15000 / 17632
[t-SNE] Computed conditional probabilities for sample 16000 / 17632
[t-SNE] Computed conditional probabilities for sample 17000 / 17632
[t-SNE] Computed conditional probabilities for sample 17632 / 17632
[t-SNE] Mean sigma: 0.178657
/opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/sklearn/manifold/_t_sne.py:986: FutureWarning: The PCA initialization in TSNE will change to have the standard deviation of PC1 equal to 1e-4 in 1.2. This will ensure better convergence.
FutureWarning,
[t-SNE] KL divergence after 250 iterations with early exaggeration: 76.980408
[t-SNE] KL divergence after 400 iterations: 2.775303
def m_embedding_norm(models):
"""Visualizes the norm and number of ratings of the movie embeddings.
Args:
model: A MFModel object.
"""
if not isinstance(models, list):
models = [models]
df = pd.DataFrame({
'title': artists.sort_values('artistID', ascending=True)['name'].values,
'num_ratings': user_artists[['artistID','userID']].sort_values('artistID', ascending=True).groupby('artistID').count()['userID'].values,
})
charts = []
brush = alt.selection_interval()
for i, model in enumerate(models):
norm_key = 'norm'+str(i)
df[norm_key] = np.linalg.norm(model.embeddings["artistID"], axis=1)
nearest = alt.selection(
type='single', encodings=['x', 'y'], on='mouseover', nearest=True,
empty='none')
base = alt.Chart().mark_circle().encode(
x='num_ratings',
y=norm_key,
color=alt.condition(brush, alt.value('#4c78a8'), alt.value('lightgray'))
).properties(
selection=nearest).add_selection(brush)
text = alt.Chart().mark_text(align='center', dx=5, dy=-5).encode(
x='num_ratings', y=norm_key,
text=alt.condition(nearest, 'title', alt.value('')))
charts.append(alt.layer(base, text))
return alt.hconcat(*charts, data=df)
Demo¶
You can find the most similar artist to a specified artist (that is contained in Last.FM) using the artist_neighbours() function. Similarily, you can find the top 10 recommendations of a particular userID [0 to 1891] using the user_recommendations() function. The first argument specifies the desired model, second argument the userID and third the top-k recommendations. Fourth argument represents the similarity measure, either DOT or COSINE (default = DOT, not a string).
user_recommendations(reg_model, 234, 10, COSINE)
| score | name | most assigned tag | |
|---|---|---|---|
| 126400 | 0.924 | The Vibrators | punk |
| 126554 | 0.923 | Moreira da Silva | N/A |
| 126490 | 0.913 | Street Bulldogs | N/A |
| 126492 | 0.912 | Pata de Elefante | rock |
| 126436 | 0.860 | The Modern Lovers | punk |
| 126397 | 0.859 | Ratos de Porão | punk |
| 126505 | 0.849 | Planet Hemp | rock |
| 115347 | 0.848 | Chico Science & Nação Zumbi | alternative |
| 126583 | 0.845 | The Saints | punk |
| 126536 | 0.842 | S.O.D. | thrash metal |
To further demonstrate the robustness of the system and measure the serendipity of our model, we incorporate the top artists that we listen to on Spotify (i.e. an unknown user). Note, these artists have to also be in the Last.FM dataset. The recommendation system should output similar artists based on it’s artist embeddings. The Spotipy library is used to interact with Spotify’s API. The similarity measure used is the Dot product. Due to the short lived nature of the spotify token and the fact you have to sign into a pop-up to retrieve the authentication token, we simply list our top 5 artists manually. If we did not, jupyter book will stall when attempting to build as it is waiting for our response. However, we provide the code used to retrieve the short-lived token for verification purposes.
"""
import spotipy
from spotipy.oauth2 import SpotifyOAuth
client_id = <insert_your_client_id>
client_secret = <insert your client secret>
redirect_url = '<insert your redirect uri>
scope = "user-top-read user-read-playback-state streaming ugc-image-upload playlist-modify-public"
authenticate_manager = spotipy.oauth2.SpotifyOAuth(client_id = client_id,client_secret = client_secret,redirect_uri =redirect_url,scope =scope,show_dialog = True)
sp = spotipy.Spotify(auth_manager=authenticate_manager)
artists_long = sp.current_user_top_artists(limit=5, time_range="long_term")
"""
top_5_artists =[
'Coldplay',
'Paramore',
'Arctic Monkeys',
'Lily Allen',
'Miley Cyrus'
]
spotify_reccomdations_df = pd.DataFrame()
for artist in top_5_artists:
similar_artist_df = artist_neighbors(reg_model, artist)[['name','dot score']]
spotify_reccomdations_df = pd.concat([spotify_reccomdations_df, similar_artist_df])
spotify_reccomdations_df.sort_values('dot score', ascending=False).head(10)
Nearest neighbors of : Coldplay.
[Found more than one matching artist. Other candidates: Jay-Z & Coldplay, Coldplay/U2]
Nearest neighbors of : Paramore.
[Found more than one matching artist. Other candidates: Paramore攀]
Nearest neighbors of : Arctic Monkeys.
[Found more than one matching artist. Other candidates: Arctic Monkeys vs The Killers]
Nearest neighbors of : Lily Allen.
Nearest neighbors of : Miley Cyrus.
[Found more than one matching artist. Other candidates: Miley Cyrus攀, Demi Lovato Ft. Miley Cyrus Ft. Selena Gomez Ft. Jonas Brothers, Miley Cyrus and Billy Ray Cyrus, Miley Cyrus and John Travolta, Hannah Montana and Miley Cyrus]
| name | dot score | |
|---|---|---|
| 3259 | Coldplay | 3.643 |
| 37842 | Paramore | 3.624 |
| 36290 | Eminem | 3.596 |
| 12363 | Muse | 3.555 |
| 6543 | Lady Gaga | 3.521 |
| 30355 | Linkin Park | 3.510 |
| 16680 | The Beatles | 3.500 |
| 24447 | Lily Allen | 3.499 |
| 17472 | The Killers | 3.481 |
| 36290 | Eminem | 3.464 |
We believe these recommodations are good as when our model was given an artist in the top five, it actually recommended other artits in the top five.
Evaluation Code¶
This is the code needed to produce the in-depth model comparison. As we decided to use different notebooks for different models, the results of this code will be combined and explained later in the book.
## create holdout test set for each user (15 items)
user_artists = pd.read_csv('data/user_artists.dat', sep='\t')
user_ids = []
holdout_artits = []
for user_id in user_artists.userID.unique():
top_15_artists = user_artists[user_artists.userID == user_id].sort_values(by='weight').head(15).artistID.tolist()
if len(top_15_artists) == 15:
holdout_artits.append(top_15_artists)
user_ids.append(user_id)
holdout_df = pd.DataFrame(data={'userID':user_ids,'holdout_artists':holdout_artits})
holdout_df.to_csv('data/evaluation/test-set.csv',index=False)
## Finding the models vanilla, regualrised predection for each user.
def get_top_15_model_predictions(model, measure):
"""Computes the top 15 predictions for a given model
Args:
model: the name of the model
measure: a string specifying the similarity measure to be used. Can be
either DOT or COSINE.
Returns:
predicted_df a dataframe containing userIDs, their top 15 artists by the model, and the correspnding scores.
"""
artist_name_id_dict = dict(zip(artists['name'], artists['artistID']))
user_ids = []
predicted_artists = []
scores_list = []
for new_user_id, orginal_user_id in orginal_user_ids.items():
top_15_names = user_recommendations(model, new_user_id, k=15,measure=measure )['name'].values
top_15_scores = user_recommendations(model, new_user_id, k=15, measure=measure )['score'].values.tolist()
artist_ids = []
for name in top_15_names:
artist_ids.append(artist_name_id_dict[name])
predicted_artists.append(artist_ids)
user_ids.append(orginal_user_id)
scores_list.append(top_15_scores)
predicted_df = pd.DataFrame(data={'userID':user_ids,'predictions_artists':predicted_artists, 'score':scores_list })
return predicted_df
# save the recommended artits into dfs and save them to data/evaluation folder
vanilla_dot_pred= get_top_15_model_predictions(vanilla_model, measure=DOT)
vanilla_cos_pred = get_top_15_model_predictions(vanilla_model, measure=COSINE)
reg_dot_pred= get_top_15_model_predictions(reg_model, measure=DOT)
reg_cos_pred = get_top_15_model_predictions(reg_model, measure=COSINE)
vanilla_dot_pred.to_csv('data/evaluation/vannila_dot_pred.csv',index=False)
vanilla_cos_pred.to_csv('data/evaluation/vanila_cos_pred.csv',index=False)
reg_dot_pred.to_csv('data/evaluation/reg_dot_pred.csv',index=False)
reg_cos_pred.to_csv('data/evaluation/reg_cos_pred.csv',index=False)